Q.Find $AB$, if $A = \begin{bmatrix} 6 & 9 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & 6 & 0 \\ 7 & 9 & 8 \end{bmatrix}$.

Concept: Matrix Multiplication Compatibility — the number of columns in $A$ must equal the number of rows in $B$. Here $A$ is $2 \times 2$ and $B$ is $2 \times 3$, so $AB$ is defined and will be $2 \times 3$.

Step 1: Compute the first row of $AB$ by multiplying row 1 of $A$ with each column of $B$:

• Column 1: $(6)(2) + (9)(7) = 12 + 63 = 75$
• Column 2: $(6)(6) + (9)(9) = 36 + 81 = 117$
• Column 3: $(6)(0) + (9)(8) = 0 + 72 = 72$

Step 2: Compute the second row using row 2 of $A$:

• Column 1: $(2)(2) + (3)(7) = 4 + 21 = 25$
• Column 2: $(2)(6) + (3)(9) = 12 + 27 = 39$
• Column 3: $(2)(0) + (3)(8) = 0 + 24 = 24$

Step 3: Assemble the resulting $2 \times 3$ matrix.

The key idea: matrix multiplication is row‑by‑column dot products. Each entry $(i,j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$. This only works if the row length of $A$ (its number of columns) matches the column height of $B$ (its number of rows). Here both are $2$, so we are good.

Let’s walk through it step by step.

1. Check compatibility

   $A$ has shape $2 \times 2$ (2 rows, 2 columns). $B$ has shape $2 \times 3$ (2 rows, 3 columns).

   The inner dimensions (the 2’s) match, so $AB$ is defined and will be $2 \times 3$ (outer dimensions: rows of $A$, columns of $B$).

2. Set up the product matrix

   We will compute three columns, each with two entries. Label the result as $C = AB$, where

3. First column of $C$ (use column 1 of $B$)
   • $c_{11}$ = row 1 of $A$ dot column 1 of $B$:

• $c_{21}$ = row 2 of $A$ dot column 1 of $B$:

4. Second column of $C$ (use column 2 of $B$)
   • $c_{12}$ = row 1 of $A$ dot column 2 of $B$:

• $c_{22}$ = row 2 of $A$ dot column 2 of $B$:

5. Third column of $C$ (use column 3 of $B$)
   • $c_{13}$ = row 1 of $A$ dot column 3 of $B$:

• $c_{23}$ = row 2 of $A$ dot column 3 of $B$:

6. Assemble the result Putting all entries together:

Method: Multiplying two matrices (row-by-column)

Use this for any product $AB$ once you have confirmed it is defined.

Steps

Step 1: Check compatibility and the result's order.

$AB$ exists iff columns of $A$ = rows of $B$; the product is (rows of $A$) $\times$ (columns of $B$).

Step 2: Compute each entry as a dot product.

The $(i,j)$ entry of $AB$ is row $i$ of $A$ dotted with column $j$ of $B$: multiply matching terms and add.

Step 3: Assemble all entries into the product matrix.

Work column by column (or row by row) so no position is missed.

Common Mistakes

Mistake 1: Multiplying entry-by-entry (Hadamard) instead of row-by-column.

Why it's wrong: matrix multiplication is a sum of products of a row and a column, not $a_{ij}\cdot b_{ij}$. Correct approach: dot each row of $A$ with each column of $B$.

Mistake 2: Pairing a row of $A$ with a row (not a column) of $B$.

Why it's wrong: the second factor must be read down its columns. Correct approach: hold a row of $A$ fixed and run down the columns of $B$.

Mistake 3: Getting the result's order wrong.

Why it's wrong: here $A$ is $2\times2$ and $B$ is $2\times3$, so $AB$ is $2\times3$, not $2\times2$. Correct approach: outer dimensions give the order.